Museums buy artwork all the time. I have never heard of them renting works. They do accept loans, but money seems to be the issue here for your character. The owner of the artwork should have the work appraised, get a few good photos taken and then do research for art institutions that might be interested. She can either solicit the institutions herself or work through an art broker. How you want to handle these details in your story is up to you.
Just a little FYI - the best prices are paid by private collectors through private dealers (not even galleries). By offering the work to a museum your character will be getting a lower price than what she might get by going to a private collector. I assume this important since the character is following the wishes of the mother who left her the work.